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I think an hour will give you plenty of time to get from Heuston to Connolly and you will even have time for a coffee:-)

The LUAS(tram) stops right outside Heuston and if you take the one for Connolly rather than the one for The Point, it will drop you right outside the station.

If you are booking your train ticket online, if you select Connolly rather than Heuston as your final destination, your train ticket will include the LUAS and save you some time queuing up to buy it outside Heuston.

If you have limited time, take the first inbound tram from Heuston, whether its for Connolly or The Point. If its destination is The Point, get off at Busáras tram stop and walk the short distance around the corner to Connolly.

As blawcleea says, the Busaras stop is literally around the corner. I sometimes jump off a Connolly LUAS at Busaras and walk across. It is faster than waiting for the Luas to negotiate the corner and switches on the track - and the pause it has to make before going into the deadend platform.

If you have limited time, take the first inbound tram from Heuston, whether its for Connolly or The Point. If its destination is The Point, get off at Busáras tram stop and walk the short distance around the corner to Connolly.

Can you book a through ticket rather than two separate ones? That would include the cost of the LUAS (and take away the ticket buying delays). I know this doesn't work if your final destination is Belfast, but it does seem to work for Dundalk.